Belief functions and inner measures

In this note we study the relation between belief functions of Dempster-Shafer theory and inner measures induced by probability functions. In [3,4] Joe Halpern and Ron Fagin claim that these classes of functions are essentially the same, or, more precisely, that they are exactly the same in case the functions are defined on formulas rather than sets. We show that, when the functions are defined on sets only a proper subclass off the belief functions over aframe S corresponds to the class of inner measures induced by a probability measurer on some algebra, on S. However, belief functions, over S do correspond to inner measures induced by probability measures defined on algebras on refinements of S. fact that in general refinemeats of S are needed to obtain all belief functions over S is shown to be obscured by the particular way formulas are assigned probabilities or weights in [3]